Finite Group Databasebeta

← #30602894

Subgroups — #30602894

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462257C211yesmaximal, characteristic
2#21461522C211yesmaximal, characteristic
3#21461506C212yesmaximal
4C2×#16840633C212yesmaximal
5#384×D61C211yesmaximal, characteristic
6C2×D61⋊M16C2212yes—
7C2×#14519856C2212yes—
8C4⋊C4×#6923—24no—
9C4×C22⋊#6923—22no—
10#16840642—22no—
11#16840635—24no—
12#16840633C418yescomplemented
13C2×C4⋊C4×D61—22no—
14C2×C4⋊C4×D61C2211yescharacteristic
15C2×C4⋊C4×D61C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16839632C411yescharacteristic, complemented
18#16839631—22no—
19#16839631C411yescharacteristic, complemented
20#16839631C411yescharacteristic, complemented
21C61×#384C411yescharacteristic, complemented
22C2×#11758213—22no—
23D61⋊M16—22no—
24D61⋊M16—22no—
25D61⋊M16—22no—
26D61⋊M16C2×C414yes—
27C2×#11758217—22no—
28C2×D61⋊C8—22no—
29#14519856—22no—
30#14519856—22no—
31#14519856C2×C414yes—
32#14519856Q814yes—
33C2×C4⋊#6923D414yes—
34C2×C22⋊#6923—22no—
35C2×C4×#6923—22no—
36C2×C4×#6923—24no—
37(C2×#6923)⋊C4—24no—
38C22×C4×D61—21no—
39C22×C4×D61—21no—
40C22×C4×D61D412yes—
41C22×C4×D61C2311yescharacteristic
42C2×C4⋊Dic61—22no—
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×#11758196—22no—
46C2×#11758196—22no—
47C2×#11758196—21no—
48C2×#11758196D412yes—
49C4⋊C4×D61—22no—
50C4⋊C4×D61—22no—
51C4⋊C4×D61—22no—
52C4⋊C4×D61—22no—
53C4⋊C4×D61—48no—
54C2×C4×Dic61—21no—
55C2×C4×Dic61C2×C411yescharacteristic
56C42×D61—22no—
57C42×D61C2×C414yes—
58C2×C4⋊C4×C61—22no—
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C42×C61C2×C411yescharacteristic
62C23×D61C2×D411yescharacteristic
63C22×#6923—24no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61C22⋊C412yes—
67C22×C4×C61C22×C411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61C22⋊C412yes—
71C22×Dic61C2×D411yescharacteristic
72C2×C4×D61—24no—
73C2×C4×D61—24no—
74C2×C4×D61—24no—
75C2×C4×D61—22no—
76C2×C4×D61—22no—
77C2×C4×D61—44no—
78C2×C4×D61—44no—
79C2×C4×D61—41no—
80C2×C4×D61—41no—
81C2×C4×D61C22×C412yes—
82C2×C4×D61C2×D412yes—
83C2×C4×D61C2×Q812yes—
84C2×C4×D61#4112yes—
85C2×C4×D61#4112yes—
86C2×C4×D61#4112yes—
87C22⋊#6923—24no—
88C22⋊#6923—24no—
89#11758217—24no—
90#11758217—42no—
91C2×#92420—42no—
92C4⋊#6923—24no—
93C4⋊#6923C4⋊C418yescomplemented
94C4×#6923—24no—
95C4×#6923—24no—
96C4×#6923—84no—
97#11758213—24no—
98#11758213—42no—
99D61⋊C8—24no—
100D61⋊C8—42no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—44no—
104C42×C61—21no—
105C42×C61C4212yescomplemented
106C4⋊Dic61—22no—
107C4⋊Dic61—22no—
108C4⋊Dic61—44no—
109#11758196—24no—
110#11758196—44no—
111#11758196—44no—
112#11758196—42no—
113C4×Dic61—21no—
114C4×Dic61—24no—
115C4×Dic61C4212yescomplemented
116C23×C61C2×C22⋊C411yescharacteristic
117C22×D61—22no—
118C22×D61—22no—
119C22×D61C4×D412yes—
120C22×D61#12011yescharacteristic
121C22×D61C22⋊Q812yes—
122C22×D61#12311yescharacteristic
123C2×#6923—28no—
124C2×#6923—44no—
125C2×#6923—44no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—41no—
133C2×C4×C61C2×C4211yescharacteristic
134C2×C4×C61#11711yescharacteristic
135C2×C4×C61C4×D412yes—
136C2×C4×C61C4×Q812yes—
137C2×Dic61—22no—
138C2×Dic61—22no—
139C2×Dic61—22no—
140C2×Dic61—22no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—41no—
144C2×Dic61C4×D412yes—
145C2×Dic61#12112yes—
146C2×Dic61#12411yescharacteristic
147C2×Dic61C4⋊Q811yescharacteristic
148C4×D61—216no—
149C4×D61—44no—
150C4×D61—82no—
151C4×D61—82no—
152C4×D61—82no—
153C4×D61C2×C4⋊C412yes—
154C4×D61#11712yes—
155C4×D61C4×D412yes—
156C4×D61C4×Q812yes—
157C4×D61C22⋊Q812yes—
158C4×D61#12312yes—
159C4×D61#12512yes—
160C4×D61#12612yes—
161#92420—44no—
162#10891—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61C4×C22⋊C411yescharacteristic
166C22×C61#38611yescharacteristic
167C22×C61#39111yescharacteristic
168C2×D61—21no—
169C2×D61—21no—
170C2×D61—24no—
171C2×D61—24no—
172C2×D61#38512yes—
173C2×D61#38712yes—
174C2×D61#45312yes—
175C2×D61#45612yes—
176#6923—48no—
177C4×C61—24no—
178C4×C61—42no—
179C4×C61—81no—
180C4×C61C4×C4⋊C412yes—
181C4×C61#38712yes—
182Dic61—24no—
183Dic61—42no—
184Dic61—81no—
185Dic61#38812yes—
186Dic61#38912yes—
187C2×#384—611no—
188#2274—611no—
189C2×#343—611no—
190#1736—612no—
191C2×#339—612no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61#143611yescharacteristic
195C2×C61#188611yescharacteristic
196C2×C61#188711yescharacteristic
197D61—22no—
198D61#185914yes—
199C22×M16—612no—
200C2×#117—612no—
201C22×C4⋊C4—611no—
202C22×C4⋊C4—611no—
203C22×C4⋊C4—1222no—
204C22×C42—611no—
205#384—611no—
206#384—611no—
207#384—611no—
208#384—1222no—
209#384#692311yescharacteristic, complemented
210C4×C4⋊C4—1224no—
211C4×C22⋊C4—1222no—
212#343—1222no—
213#340—1224no—
214#339—618no—
215C61#1089111yescharacteristic, complemented, Sylow
216C23×C4—611no—
217C23×C4—612no—
218C23×C4—1221no—
219C23×C4—1221no—
220C2×M16—614no—
221C2×M16—1222no—
222C2×M16—1222no—
223C2×M16—1222no—
224C2×M16—1222no—
225C2×M16—1222no—
226C22×C8—1222no—
227#117—614no—
228#117—614no—
229#117—1222no—
230#117—1222no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—614no—
233C2×C4⋊C4—612no—
234C2×C4⋊C4—611no—
235C2×C4⋊C4—611no—
236C2×C4⋊C4—1222no—
237C2×C4⋊C4—1222no—
238C2×C4⋊C4—1222no—
239C2×C4⋊C4—1222no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4—1222no—
242C2×C4⋊C4—1222no—
243C2×C4⋊C4—1222no—
244C2×C4⋊C4—2448no—
245C2×C4⋊C4C2×#692311yescharacteristic
246C2×C4⋊C4C2×#692311yescharacteristic
247C2×C22⋊C4—1222no—
248C2×C42—611no—
249C2×C42—614no—
250C2×C42—1222no—
251C2×C42—1221no—
252C2×C42—1222no—
253C2×C42—1224no—
254C2×C42C2×#692311yescharacteristic
255#95—1224no—
256C24—611no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—611no—
260C22×C4—612no—
261C22×C4—612no—
262C22×C4—612no—
263C22×C4—612no—
264C22×C4—612no—
265C22×C4—612no—
266C22×C4—612no—
267C22×C4—1224no—
268C22×C4—1224no—
269C22×C4—1224no—
270C22×C4—1222no—
271C22×C4—1222no—
272C22×C4—1221no—
273C22×C4—1224no—
274C22×C4—1221no—
275C22×C4—2441no—
276C22×C4—2441no—
277C22×C4—2444no—
278C22×C4—2444no—
279C22×C4C22⋊#692312yes—
280C22×C4C22×#692311yescharacteristic
281M16—1224no—
282M16—1224no—
283M16—2442no—
284M16—2442no—
285C2×C8—1224no—
286C2×C8—2442no—
287C2×C8—2442no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—44no—
291C4⋊C4—618no—
292C4⋊C4—1224no—
293C4⋊C4—1224no—
294C4⋊C4—1222no—
295C4⋊C4—1222no—
296C4⋊C4—2444no—
297C4⋊C4—2442no—
298C4⋊C4—2444no—
299C4⋊C4—2444no—
300C22⋊C4—1224no—
301C22⋊C4—1224no—
302C42—21no—
303C42—612no—
304C42—1224no—
305C42—1224no—
306C42—1224no—
307C42—1221no—
308C42—4884no—
309C42C4×#692312yescomplemented
310C23—611no—
311C23—611no—
312C23—612no—
313C23—612no—
314C23—1222no—
315C23—1222no—
316C23C2×C22⋊#692311yescharacteristic
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—612no—
325C2×C4—611no—
326C2×C4—611no—
327C2×C4—612no—
328C2×C4—612no—
329C2×C4—612no—
330C2×C4—612no—
331C2×C4—612no—
332C2×C4—612no—
333C2×C4—612no—
334C2×C4—612no—
335C2×C4—612no—
336C2×C4—12216no—
337C2×C4—1222no—
338C2×C4—1222no—
339C2×C4—1222no—
340C2×C4—1228no—
341C2×C4—1222no—
342C2×C4—4882no—
343C2×C4—4882no—
344C2×C4—4882no—
345C2×C4—2444no—
346C2×C4—2442no—
347C2×C4—2441no—
348C2×C4—2444no—
349C2×C4—2444no—
350C2×C4—2442no—
351C2×C4C2×C4×#692311yescharacteristic
352C2×C4#1451985611yescharacteristic
353C2×C4D4×#692312yes—
354C2×C4Q8×#692312yes—
355C8—2444no—
356C22—21no—
357C22—21no—
358C22—612no—
359C22—612no—
360C22—612no—
361C22—612no—
362C22—1224no—
363C22—1224no—
364C22—1221no—
365C22—1221no—
366C22C4×C22⋊#692311yescharacteristic
367C22#1684071011yescharacteristic
368C22(C2×#6923)⋊Q811yescharacteristic
369C4—24no—
370C4—42no—
371C4—81no—
372C4—612no—
373C4—612no—
374C4—1224no—
375C4—4881no—
376C4—2448no—
377C4—2442no—
378C4C4⋊C4×#692312yes—
379C4#1684071112yes—
380C2—21no—
381C2—21no—
382C2—614no—
383C2—1222no—
384C2#2146099811yescharacteristic
385C2#2146182511yescharacteristic
386C2#2146182611yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#384#6923
hypocenterC61#10891
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. #16840633C4semidirect
17. #16839632C4semidirect
19. #16839631C4semidirect
20. #16839631C4semidirect
21. C61×#384C4semidirect
93. C4⋊#6923C4⋊C4semidirect
105. C42×C61C42semidirect
115. C4×Dic61C42semidirect
209. #384#6923semidirect
215. C61#10891semidirect
309. C42C4×#6923semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214622572
#214615222
#214615062
C2×#168406332
#384×D612
#1089161